Solve for $m$: $(m-4)^3 = \left(\frac 18\right)^{-1}$.
We have $\left(\frac{1}{8}\right)^{-1}=8=2^3$, so we can write the given equation as $$(m-4)^3=2^3.$$ Therefore, $m-4 = 2$, so $m=\boxed{6}$.